The equations for the coupled system of inflaton, radiation and metric perturbations can be found in [1]}, [2]}. At linear order, the FLRW metric is given by:
\(ds^2= - (1 + 2 \alpha ) dt^2 -2 a \partial _i \beta dx^i dt + a^2(t)[ \delta _{ij} ( 1 + 2 \varphi )+2 \partial _i \partial _j \gamma ] dx^i dx^j \,.\) 
